Firms, Trade and Labor Market Dynamics Oleg Itskhoki Itskhoki@Princeton.edu Elhanan Helpman EHelpman@Harvard.edu June 29, 2014 Work in progress: preliminary and incomplete Download the most up-to-date version from: http://www.princeton.edu/~itskhoki/papers/tradelmdynamics.pdf Abstract Adjustment to trade liberalization is associated with substantial reallocation of labor across firms within sectors. This salient feature of the data is well captured by the new generation of trade models, which however assume frictionless and instantaneous adjustments in the labor market. A natural question is whether labor market frictions slow down this reallocation process, thereby dissipating the gains from trade along the transition path. In this paper, we develop a model with heterogeneous firms and Diamond-Mortensen-Pissarides type frictions in the labor market, in which we fully characterize the transitional dynamic response to a trade liberalization. The sunk cost of hiring workers makes low-productivity non-exporting firms reluctant to fire workers and exit in the short run, which in turn crowds out the new more-productive entrants. This depresses aggregate productivity and trade flows during the adjustment process. Yet, despite the lengthy dynamic adjustment, the consumer gains from trade in this economy are achieved instantaneously and do not depend on the extent of the labor market frictions. Lastly, the trade shock creates short-run winners and losers among the ex ante homogenous workers, with bad jobs concentrated in the non-exporting firms adversely affected by trade liberalization. We thank Felix Tintelnot, Steve Redding, Richard Rogerson, Esteban Rossi-Hansberg, Ezra Oberfield, David Weinstein, Andrew Bernard and seminar participants at Princeton, UBC, West Coast Trade Workshop at UCLA, ERWIT at the University of Oslo for useful comments, and Ricardo Reyes-Heroles for excellent research assistance.
1 Introduction Adjustment to trade liberalization is associated with substantial reallocation of labor, both across sectors, but even more importantly across firms within sectors (see e.g. Levinson, 1999). The new generation of trade models, following Melitz (2003), captures well this salient feature of the data, predicting large post-trade-liberalization reallocation of workers from shrinking non-exporting firms facing increased export competition towards expanding exporting firms (see e.g. Eaton, Kortum, and Kramarz, 2011). Much of the analysis in this literature, however, is confined either to the steady state comparative statics, or to the case of the frictionless labor markets, in which labor transitions happen instantaneously and all workers benefit from trade independently of their employment history. A natural question, then, is to what extent labor market frictions slow down this reallocation process across firms, depressing the productivity and reducing the gains from trade in the short run, during the transition period. Additionally, do labor market frictions result in unequal distributional consequences for labor market outcomes, in particular creating winners and losers from trade in an ex ante homogenous pool of workers? We address this questions by developing a tractable trade model with heterogeneous firms and Diamond-Mortensen- Pissarides (DMP) search frictions in the labor market (as described in e.g. Pissarides, 2000), in which we fully characterize the transitional dynamics in response to a trade liberalization. The model features two symmetric countries with a non-traded and a traded sector, both subject to the same labor market frictions. Firms in the traded sector are large (multiworker), heterogenous, monopolistic competitors facing both fixed and variable trade costs to access the foreign market, as in Melitz (2003). There is free entry of firms in both sectors, and upon entry the firms can post costly vacancies to attract workers. The unemployed workers are perfectly mobile across sectors and, upon choosing a sector, can search for a job and randomly meet vacant firms. Upon matching, a firm and its workers bargain about wages without commitment. In order to switch jobs, both within or across sectors, workers need to first separate into unemployment. In this economy, we study the dynamic response to an unanticipated bilateral reduction in the trade costs. In the new steady state, the least productive of the previously active firms exit, the firms in the middle of the productivity distribution shrink their employment under pressure from foreign competition, while the most productive firms expand their export sales and employment. This leads to a productivity and welfare improvement economy-wide and for each and every worker. We show, however, that the transition dynamics to this new steady state can take a long time, with less productive incumbent firms reluctant to fire their workers and exit, and choosing instead to gradually shrink their employment subject to the natural attrition forces. Continued employment and sales by these firms crowds out the new more productive entrants. 1
This results in misallocation of labor across firms, which in turn reduces the productivity in the traded sector. 1 Furthermore, as non-exporters are relatively more prevalent among the incumbents, international trade flows are depressed during the transition, and reach their new steady state level only gradually, as less productive incumbents are replaced by a more selected group of new entrants. This labor market dynamics creates distributional consequences for workers, which are tied to the heterogeneous outcomes of their employers in response to the trade liberalization. Workers employed by expanding firms, as well as the unemployed workers, gain from trade. At the same time, workers employed by less productive incumbents, which need to shrink after trade liberalization, fare less well and may even loose from trade. Some of these workers have to experience a spell of unemployment if their firm exits or fires part of its labor force, however, the other such workers see a decline in their wages during the transition period. 2 Therefore, the interaction between firm heterogeneity and labor market frictions creates good and bad jobs during the transition in response to a trade liberalization. These heterogenous outcomes for observationally identical workers, tied to the outcomes of their employers after a trade liberalization episode are consistent with the recent empirical evidence (see Verhoogen, 2008; Amiti and Davis, 2011; Helpman, Itskhoki, Muendler, and Redding, 2012). Despite these rich cross-sectional patters during the potentially lengthy transition, the consumer gains from trade are realized instantaneously and, furthermore, do not depend on the extent of labor market frictions. Both of these results are intriguing. We show that the proportional long-run gains from trade are equal to the gains in the consumer surplus, which do not depend on the labor market frictions, although these frictions are important in determining the long-run levels of productivity and welfare (as well as comparative advantage, as in the static model of Helpman and Itskhoki, 2010). The fact that the consumer gains from trade are realized instantaneously relies on two main assumptions that we adopt the free entry condition for firms and the mobility of unemployed workers across the traded and non-trade sector. The mobility of workers ties down the labor market conditions in the traded sector to those in the rest of the economy, assuming the traded sector never exhausts the full economy-wide pool of unemployed. The free entry condition in the non-traded sector determines the labor market conditions, while the free entry condition in the traded sector ties down the product market competition to the 1 There is a second force affecting the sectoral productivity, namely the positive variety effect from the incumbents surviving in the short run (as emphasized, for example, in Alessandria, Choi, and Ruhl, 2013). We show that the misallocation effect dominates in the short run, resulting in reduced productivity, while the variety effect dominates towards the end of the transition, when the unproductive incumbent firms shrink below a small enough size. 2 The income losses for these workers are, of course, bounded above by their outside option of separating into unemployment and finding a new job. 2
labor market conditions and trade costs to ensure expected zero profits for the new entrants. Under these circumstances, there exists a unique level of prices (and hence of the consumer surplus), which is consistent with the new lower level of trade costs throughout the whole transition period. This free-entry logic is similar to that in Atkeson and Burstein (2010) in the context of a model with technology adoption and a frictionless labor market. The overall dynamic gains from trade in our model equal the difference between the consumer gains from trade and the household income loss due to the reduction in worker wages and firm profits during the transition period. We show quantitatively that the aggregate income loss component, although increasing in the extent of labor market frictions, is small relative to the consumer gains from trade. At the same time, since the decline in wage income is heterogeneous across workers, it can lead to considerable distributional consequence. We further show that the balance of income losses between firms and workers depends on the extent of labor market frictions. In particular, firms bear most of the losses in rigid labor markets due to the large sunk hiring costs, which they have incurred before the trade shock and which make them reluctant to destroy the matches and fire their employees. By consequence, beyond a certain level of labor market rigidity, further increase in labor market frictions shields workers from separation into unemployment and short-run income losses after a trade shock. To make our model tractable, we adopt a number of strong assumptions, in particular by make the model linear along various dimensions, including firm entry and hiring costs. We view this as a useful environment to clearly isolate the qualitative forces shaping the dynamic heterogenous adjustment to a trade shock in the cross-section of firms, focusing on the direct effects of the labor market frictions and separating it from other mechanisms that result in a period of transitional dynamics even in the absence of any labor market rigidities. In particular, labor market frictions in the model result in hiring costs and an associated ss nature of the firm employment decisions, along with an inaction region, which is the source of the transitional dynamics in the labor market. In Section 6, we provide a detailed discussion of the assumptions and the direction in which the future quantitative work can relax them. Related Literature The model in this paper builds on our earlier work, Helpman and Itskhoki (2010), in which we study the the long-run effects of labor market and trade reforms in countries with asymmetric labor market institutions and heterogenous firms. 3 consequences of labor market frictions for transition dynamics in neoclassical trade models 3 Felbermayr, Prat, and Schmerer (2011) and others also study the steady state effects of a trade liberalization in an economy with heterogenous firms and search frictions in the labor market; Davis and Harrigan (2011) and Egger and Kreickemeier (2009) analyze the effects of other labor market frictions in a trade model with heterogeneous firms. The 3
were studied by Davidson, Martin, and Matusz (1999), Kambourov (2009) and Coşar (2010). Labor market dynamics with heterogenous firms were analyzed in Coşar, Guner, and Tybout (2011), Fajgelbaum (2013), Danziger (2013), Cacciatore (2013) and Felbermayr, Impullitti, and Prat (2014). Our paper is the first to fully characterize the transition dynamics in an economy with heterogeneous firms in response to an aggregate trade shock. In macro-labor literature, the dynamics of labor market with heterogeneous firms were studied in Acemoglu and Hawkins (2013), Elsby and Michaels (2013) and Schaal (2012). 2 Setup Consider a world of two symmetric countries, each producing two goods a non-traded homogenous good and a traded differentiated good. The differentiated good is produced by heterogeneous firms under monopolistic competition, and exporting is associated with both variable and fixed trade costs. The labor market in each sector is subject to a random search friction with wage bargaining upon matching. We setup the model in discrete time with short time intervals, and use the continuous-time approximation to simplify notation, while the appendix provides exact discrete-time expressions. 2.1 Households Each country is populated by a unit continuum of identical infinitely-lived households with a per period utility function u ( q 0, Q ) over the consumption of a homogenous good q 0 and a differentiated good Q, and with an annualized discount rate r. We suppress the dependence on time t where it leads to no confusion. The differentiated good is a CES aggregator of individual varieties ( 1/β Q = q(ω) dω) β, 0 < β < 1, (1) ω Ω where Ω is the set of varieties ω available for consumption and ε 1/(1 β) > 1 is the elasticity of substitution between the varieties. We choose the non-traded homogenous good as numeraire setting p 0 1. 4 following functional form assumption: Assumption 1 The utility function is quasi-linear: We make the u(q 0, Q) = q 0 + 1 ζ Qζ for (q 0, Q) R R +, (2) 4 Even if the homogenous good were tradable, in equilibrium with symmetric countries it is non-traded. 4
with ζ (0, β). 5 This is a strong assumption. However, it allows to focus our analysis on the dynamic effects stemming from labor market frictions, unconfounded by the effects of the curvature in the utility function on the timing of entry of firms, which operates independently of the labor market frictions. This assumption can be equivalently replaced by an assumption of a perfectly elastic supply of capital at interest rate r, common in small-open-economy models. In either case, the assumption has a partial equilibrium flavor, which we think is a reasonable point of approximation, since only a small fraction of consumer expenditure is on tradables. Hence, a reduction in trade costs affects directly only a part of the economy, and the homogenous-good sector represents in our analysis the rest of the economy not affected directly by trade. Under Assumption 1, the flow utility can be written as: 6 u t = I t + 1 ζ Q ζ t, ζ where the first term is income and the second term is consumer surplus from the differentiated good. Therefore, the lifetime utility equals the discounted present value of household income and consumer surpluses, and trade in general affects both components of the utility. Each household has (a measure) L of workers, who are allocated between the two sectors. Workers in each sector can be either employed or unemployed and searching for a job. Unemployed workers can frictionlessly reallocate between the two sectors, while employed workers need to first separate into unemployment before starting to search for a new job in either sector. Unemployed workers in both sectors face Diamond-Mortensen-Pissarides search frictions, as we describe in detail below. The workers receive an unemployment benefit b u in units of the homogenous good while unemployed and searching for a job, which is financed by a lump sum tax on all households. 2.2 Non-traded sector A match between a firm and a worker in this sector produces a constant flow of the homogenous good per period of length. We assume that is small (e.g., = 1/12, corresponding to a period length of one month), rendering accurate the approximation around = 0. 5 This parameter restriction implies that the differentiated varieties are better substitutes with each other than with the homogenous good. 6 With this utility function, the market clearing interest rate equals discount rate, r, and hence expenditure in the indirect utility function can be assumed to equal income without loss of generality. Denote by P the price index of the differentiated good in units of the numeraire. A period s expenditure equals q 0 + P Q = I. The optimal choice of Q satisfies P Q = Q ζ, and therefore the period s flow utility is I P Q + Q ζ /ζ = I + 1 ζ ζ Qζ. 5
The market for homogenous goods is competitive. A firm can enter this sector freely, and post costly vacancies to attract workers. We denote the expected cost to a firm of attracting a worker by b 0, equal to the cost of a vacancy divided by the vacancy-filling rate. In the appendix we show that assuming a Cobb-Douglas matching function between sectoral vacancies and unemployed searching for work in the sector, we have the following relationship: b 0 = a 0 x α 0, (3) where a 0 is a derived parameter increasing in the cost of a vacancy and decreasing in the productivity of the matching technology, and x 0 is the job finding rate a measure of the sectoral labor market tightness. 7 Upon matching, the firm and the worker Nash-bargain over the wage rate without commitment. A match is exogenously destroyed with a constant hazard rate s 0. We make the following assumption: Assumption 2 The homogenous good sector is large enough relative to the differentiated good sector (i.e., L is large enough), that along the equilibrium path the stock of unemployed searching for a job in the homogenous-good sector is positive, U 0,t > 0, in every period t. With this assumption, in the appendix we prove the following proposition which characterizes the labor market equilibrium in the homogenous sector: Lemma 1 Under Assumption 2, the job finding rate x 0 and the hiring cost b 0 are positive, finite, constant over time, and solve [ ] 2(r + s0 ) + x 0 b0 = 1 b u (4) together with (3); and the economy-wide value to an unemployed worker is constant over time and given by: rj U 0 = b u + x 0 b 0. (5) Since unemployed are mobile across sectors, and given that some of the unemployed always search for work in the homogenous sector (Assumption 2), J0 U in Lemma 1 characterizes the economy-wide value to unemployed from the partial equilibrium in the homogenous sector s labor market. The driving force behind this result is the free entry condition for firms in the homogenous sector, which equalizes the value of a filled vacancy, J F 0, with the hiring cost, b 0. This, in turn, is only consistent with a single value of the labor market tightness, 7 Indeed, with a Cobb-Douglas matching function, the job finding rate is a power function of the vacancyunemployment ratio a conventional measure of the labor market tightness. 6
x 0, characterized by (3) (4). Nash bargaining equalizes the surplus from the employment relationship between the firm and the worker. As a result, an unemployed worker receives b u and expects to find employment at rate x 0 with the surplus from employment given by b 0, as reflected in (5). In addition, we show that the equilibrium wage rate in the homogenous sector equals: w 0 = b u + (r + s 0 + x 0 )b 0. (6) Assumption 2 ensures that there always are unemployed workers searching for a homogenous-sector job, and therefore firms always enter and post vacancies in this sector. In other words, we require that the trade shock to the differentiated sector is never large enough to lead this sector to absorb all economy-wide unemployment, even for a single period. This arguably is a realistic assumption in view of the modest employment share of the tradable sector. By pinning down the economy-wide value to unemployed, Assumption 2 and Lemma 1 ensure a useful block-recursive structure of the model. In Section 4, we evaluate this assumption quantitatively, and relax it in Section 6. 2.3 Traded sector A firm in the differentiated sector pays a sunk cost f e in terms of the numeraire (homogenous good) to enter the industry with a unique variety ω and draws its productivity θ from a known distribution G(θ). To simplify exposition, we adopt the assumption that the productivity distribution is Pareto, G(θ) = 1 θ k, with the shape parameter k > ε 1, but this is not required for our qualitative results. The firm then has access to a linear technology producing y = θh units of its variety when it employs h workers. The firm faces flow fixed cost of operation f d, and it can additionally choose to export its variety by paying a flow fixed exporting cost f x. All fixed costs are in terms of the numeraire. Exports are additionally subject to an iceberg variable trade cost τ 1, i.e. τ units of a good must be shipped out in order for one unit of the good to arrive to the foreign market. We show in the appendix that, given the CES preference aggregator (1), the revenue function of a firm with productivity θ is given by: R(h, ι; θ) = Θ(ι; θ) h β, Θ(ι; θ) [ 1 + ιτ β ( Q Q ) β ζ ] Q β ζ β θ, (7) where ι {0, 1} is an indicator of whether the firm exports, while Q and Q characterize the product market competition at home and in foreign. Note that the revenue of a nonexporting firm is simply Q (β ζ) y β, while an exporting firm (with ι = 1) optimally splits its output y = θh between the domestic and foreign markets, which results in its revenues shifted 7
outwards, as reflected by the square bracket in (7). The firms have to pay a cost of b units of the numeraire to hire one unit (measure one) of workers. Similarly to (3), the Cobb-Douglas matching function links this hiring cost to the job finding rate of workers in this sector, x: b = ax α. (8) Upon matching the firm bargains with its workers (without commitment) according to Stole and Zwiebel (1996). That is, the firm bargains bilaterally with each of its workers according to Nash, taking into account that the departure of the worker will cause a renegotiation of wages for all of its remaining workers. The bargaining is over the revenues of the firm once the employment decision and the per-period fixed production and exporting costs are sunk. In view of Lemma 1 and given the functional form of the revenue function in (7), we prove in the appendix the following result: Lemma 2 The outcome of the bargaining game between a firm and its h workers is the wage schedule w(h, ι; θ) = β R(h, ι; θ) + 1 1 + β h 2 rj U, (9) where the value to unemployed J U = J U 0 is characterized in Lemma 1. Importantly, the wage schedule in Lemma 2 applies equally to both firms that expand and reduce their labor force, and independently of whether h is the optimal level of employment of the firm. The bargained wage rate in (9) partly compensates the workers for the forgone flow value of unemployment (rj U ) and in addition delivers them their share in surplus equal to a constant fraction of average revenues of the firm. Combining together (9) and (7), we can write the flow operating profit of the firm, gross of hiring costs, as: ϕ(h, ι; θ) = 1 1 + β R(h, ι; θ) 1 2 rj U h f d ιf x. (10) A firm exogenously separates with a fraction of its workforce at an annualized rate σ and it dies at an annualized rate δ. We denote by s = σ + δ the overall exogenous rate for a worker employed in the differentiated sector to be separated into unemployment. A firm can fire some or all of its workers at no cost. Therefore, a firm needs to pay the hiring cost of C(h, h) = b max { h (1 σ )h, 0 } (11) to change its employment from h to h next period. Part of the hiring cost (σh ) is borne to 8
replace the exogenous labor force attrition, and the rest (h h) is payed to increase the size of the labor force, while non-hiring firms (with h (1 σ )h) incur no costs. With this setup, we characterize the labor market equilibrium in the differentiated sector: Lemma 3 (a) The job finding rate x and the hiring cost b in the differentiated sector are constant, and satisfy xb = x 0 b 0 ; (b) The optimal employment of a hiring firm is given by h(ι; θ) = Φ 1/β Θ(ι; θ), where Φ ( ) β 2β 1 1 + β b u + [ 2(r + s) + x ] b (12) and Θ(ι; θ) is defined in (7). The formal proof of this lemma is presented in the appendix, and here we describe the logic behind this result. 8 A hiring firm equalizes the value from its marginal worker with the cost of hiring, b. The splitting of the surplus in bargaining, in turn, ensures that the employment value to the workers equals the employment value to the firm. Therefore, the unemployed workers in the differentiated sector have the job finding rate of x and the gain in value of b upon employment, in comparison to, correspondingly, x 0 and b 0 in the homogenous sector (see Lemma 1). The indifference of unemployed workers between the two sectors then requires xb = x 0 b 0, which in view of (8) results in: x = x 0 ( a a 0 ) 1 1+α and b = b 0 ( a a 0 ) 1 1+α, (13) where a/a 0 is the inverse measure of the efficiency of the matching technology in the differentiated sector relative to the homogenous sector. 8 The proof uses the recursive Bellman equation for a firm with productivity θ: { J F (h) = max ϕ(h) C(h, h) + 1 δ } h 1 + r J + F (h ) where J F ( ) and J+ F ( ) denote the current and next period value functions of the firm. The first order condition for the choice of h is: 1 δ 1 + r J F h,+(h ) = and, making use of it, the Envelope Theorem can be written as: 0, if h < (1 σ )h, [0, b], if h = (1 σ )h, b, if h > (1 σ )h, J F h (h) = ϕ (h) + 1 s 1 + r J F h,+(h ), where the subscript h indicates the partial derivative with respect to employment. The inequalities in the first order condition reflect the ss nature of the labor force adjustment in this model. Provided a constant b, we have Jh F = J h,+ F = 1+r 1 δ b for a hiring firm, which together with the Envelope Theorem characterizes optimal employment, ϕ (h) = r+s 1 δ b. The approximation with 0 yields J h F = b and ϕ (h) = (r + s)b. 9
The optimal employment rule (12) in the second part of Lemma 3 results from the equalization of the flow value from the marginal workers, ϕ (h), characterized by (10), with the flow cost of hiring an extra worker, (r + s)b. The derived parameters Φ is the summary statistic for the extent of labor market imperfections, and it decreases in the hiring cost b. Indeed, more productive firms and exporters have larger optimal employment (due to higher Θ(ι; θ)), while all firms are smaller in a more frictional labor market (due to lower Φ). 9 Note that the employment of a firm is a jump variable. Due to the linearity of the hiring cost, the firm immediately jumps its employment up to the optimal level, and in a stationary environment the firm then maintains this employment level each period by hiring to exactly offset the attrition. At the same time, if the firm has more workers relative to the optimum, it does not fire them immediately, and waits until its labor force shrinks as a result of the exogenous attrition. Substituting the optimal employment level (12) into the wage schedule (9), we obtain the equilibrium wage rate paid to all employed works by the hiring firms: w = b u + (r + s + x)b, (14) which parallels expressions (6) for the wage rate in the homogenous sector. Note that all hiring firms in this economy pay the same wages, independently of their productivity, by adjusting their employment on the extensive margin. The non-hiring firms and firing firms, in contrast, pay a lower wage rate, as their employment is above the optimal level. 10 Lastly, we introduce notation J V (θ) for the value function of a firm with productivity θ and zero workforce in a given time period, which allows us to write the free entry condition as J V (θ)dg(θ) f e, (15) which holds with equality when there is entry of firms at that time period. Firms with V (θ) 0 hire workers and produce starting next period, while firms with V (θ) < 0 exit immediately. The following result provides a sharp explicit expression for the value of the 9 Helpman and Itskhoki (2010) focus on the effects of cross-country differences in labor market frictions (Φ) on the steady state comparative advantage and asymmetric gains from trade. Note that using (4) and Lemma 3(a), one can show that Φ is decreasing in (sb s 0 b 0 ). Therefore, countries with higher overall level of labor market frictions (high b = b 0 ) have comparative advantage in sectors with greater labor market turnover (higher relative separation rate, s/s 0 ), consistent with the evidence in Cuñat and Melitz (2011). 10 Indeed, from (9) and (7), the wage rate is a decreasing function of employment, and a firm chooses to reduce its labor force only if its current employment exceeds the desired level given by (12) and resulting in wage rate in (14). Furthermore, the wage rate paid by firing firms equals the flow value of unemployment, rj U = b u + xb, as employment in this case yields no surplus (i.e., Jh F = J E (h) J U = 0). Therefore, wages paid by non-hiring firms fall within the range from b u + xb paid by firing firms to b u + (r + s + x)b paid by hiring firms. 10
firm in one special case, which turns out particularly relevant for our analysis: 11 Lemma 4 The value of a firm with productivity θ and zero employees, which hires workers in every future period, solves the following difference equation: { } (r + δ)j 1(θ) V J 1 β V (θ) = max ι {0,1} 1 + β ΦΘ(ι; θ) f d ιf x, (16) where the ( 1)-subscript denotes the previous period, J V (θ) (J V (θ) J V (θ))/, and Θ(ι; θ) and Φ are defined in (7) and (12) respectively. Lemma 4 applies for a general time path of aggregate state variables such as b and Q, which affect Φ and Θ in (16). Furthermore, the reason the requirement of Lemma 4 that a firm hires in every future period is not overly restrictive, as it is satisfied, for example, for firms that simply maintain their employment level by hiring to offset attrition. Finally, equation (16) is derived under the assumption that the exporting fixed cost is not sunk, and a firm can choose whether to pay a fixed cost and export every period. We adopt this assumption to simplify exposition, but it is not necessary for our results. To safe space, we provide the formal conditions for production and exporting under the special circumstances discussed below. 2.4 General equilibrium To close the model, we need to characterize the aggregate employment and number of firms in the differentiated sector. We denote by M the number (measure) of firms that have entered the differentiated sector, and did not die yet for an exogenous reason. Therefore, M evolves according to: M + = (1 δ )M + M e, where M e is the number of entrants in a given period. M e 0 holds with complementary slackness with the free entry condition (15). Note that the number of firms is a jump variable, an assumption that we relax in Section 6. Further denote by G(h, θ) the joint cumulative distribution function of firm employment and productivity in a give time period among the M currently active firms. The M e new entrants have zero employment until the following period. The aggregate employment in the differentiated sector is then: H = M hdg(h, θ), (17) 11 The formal proof of this lemma is in the appendix, and it combines the Bellman equation for the value function of the firm with the optimal employment policy function of a hiring firm described in footnote 8. 11
The evolution of G( ) is characterized by the firm s employment policy functions described in footnote 8. Given the number of firms and their employment and exporting decisions we can use (1) to compute the consumption of the differentiated good, Q. Finally, this also allows us to recover the aggregate number of vacancies posted in the differentiated sector V, and the corresponding sectoral unemployment U in order to maintain the constant labor market tightness x given in (13). This determines the number of workers attached to the two sectors in a given period, N = H + U and N 0 = L N. We provide further details in the appendix. 3 Long-run Equilibrium Consider a symmetric steady state in which all variables have the same values at home and in foreign, and in particular Q = Q. Due to the positive exogenous workforce attrition rate (σ > 0), all producing firms in steady state hire workers in every period to offset attrition, and therefore their employment is given by (12). Due to the positive firm death rate (δ > 0), there is constant firm entry in steady state, and therefore the free entry condition holds with equality. Now consider an entrant in a steady state environment. The conditions of Lemma 4 are satisfied for all firms that do not exit immediately, and therefore the steady-state value of an entrant with productivity θ and zero employment is given by J V (θ) = 1 { } 1 β r + δ max ι {0,1} 1 + β ΦΘ(ι; θ) f d ιf x. (18) The solution to the maximization problem in (18) defined the exporting cutoff θ x such that ι(θ) 1 {θ θx}. We also define the production cutoffs, θ d, from J V (θ d ) = 0, such that all firms with θ θ d hire workers and produce in the long run, while firms with θ < θ d exit the industry immediately upon entry. Using the definition of Θ(ι; θ) in (7), the two long-run cutoff conditions can be written as (see the appendix): β ζ ΦQ 1+β θ ε 1 d = f d, (19) θ x /θ d = τ (f x /f d ) 1/(ε 1), (20) and we choose the value of the fixed cost f x large enough that θ x > θ d. These cutoff conditions, together with the free entry condition (15), allow us to solve for the long-run values of 12
(θ d, θ x, Q) as functions of the hiring cost b and trade costs τ. 12 Next, using (1) and (17), we can solve for the steady state employment in the differentiated sector (see the appendix): H = Φ β Q ζ, (21) and, additionally using the Pareto productivity assumption, the number of firms M in the differentiated sector: M = 1 k(r + δ)f e β 1 + β Qζ. (22) Both aggregate sectoral employment and the number of firms are proportional to Q ζ, which equals total sales of the differentiated good. Finally, note that sh is the total number of hires in the differentiated sector each period (σh to replace attrition and δh by new entrants). In steady state the flow in and out of unemployment are equalized, and hence sh = xu. Given the value of x, this pins the total number of workers assigned to the two sectors in the economy: N = (1 + s/x)h and N 0 = L N. This is sufficient to recover the remaining equilibrium variables, in particular the output of the homogenous good and the amount of the homogenous good spent on fixed (entry, production, exports) and hiring costs. The production of the homogenous good is a linear function of the size of the economy, determined by L, and we require that L is large enough to ensure positive consumption of the outside good. Since firms make zero profits on average, we measure the steady state welfare as the sum of the consumer surplus from the differentiated good, (1 ζ)q ζ /ζ, and the employment income, I = w 0 H 0 + wh. 13 3.1 Long-run gains from trade Given the equilibrium conditions described above, we can immediately prove the following comparative statics result across steady states: 12 Using (19) (20) and (18), the free entry condition (15) can be simplified to: f d θ θ d [ (θ/θd ) ε 1 1 ] dg(θ) + f x θ θ x [ (θ/θx ) ε 1 1 ] dg(θ) = (r + δ)f e, [ 1 which under the Pareto distribution further simplifies to k/(ε 1) 1 fd θ k d ] + f x θx k = (r + δ)fe. Note that all steady state equations closely parallel their analogs in the static model in Helpman and Itskhoki (2010). 13 In case with symmetric labor market frictions in both sectors (s = s 0 and a = a 0 ), both the wage rate and the unemployment rate in both sectors are the same and equal to w = b u + (r + s + x)b and u = s/(x + s) respectively. As a result, labor market income equals I = w xl/(x + s), where the second term is the economy-wide employment. Finally, since the expenditure on the differentiated good is equal to Q ζ, it is sufficient to require I > Q ζ, or equivalently L > (1 + s/x) Q ζ /w, to ensure positive consumption of the outside good in steady state. 13
Proposition 1 In a symmetric world economy, a bilateral reduction in trade costs leads to: (a) an increase in Q, H and M, with the proportional changes in these variables independent of the extent of labor market frictions; (b) assuming additionally symmetric labor markets across sectors (s = s 0 and a = a 0 ), the aggregate unemployment and labor income in terms of the homogenous good do not change with trade costs, and the steady state welfare gains from trade do not depend on the extent of the labor market frictions. Proof: The export cutoff condition (20) and the free entry condition in footnote 12 determine the production and export cutoffs θ d and θ x as functions of the product market parameters and trade costs only, and independently from the labor market parameters. A reduction in τ results in an increase in θ d. Further, from the production cutoff condition (19) and equations (21) and (22), we immediately have: 14 ( Q where prime denotes the new steady state. Q ) ζ = H H = M M = ( θ d θ d ) βζ β ζ, (23) Next, from Lemmas 1 and 3, the labor market outcomes x 0, x, b 0 and b do not depend on the trade costs, and therefore the reduction in trade costs leaves unchanged the labor market outcomes. With symmetric labor markets, we have x 0 = x and b 0 = b (see (13)), and the wage rates are equalized across all workers in both sectors, w 0 = w = b u + (r + s + x)b (see (6) and (14)), both before and after reduction in trade costs. We adopt the following measure of the welfare gains from trade: [ GT (I I) + 1 ζ ζ (Q ) ζ Q ζ], (24) 1 ζ ζ Qζ i.e. the change in the consumer surplus plus the change in market income relative to the initial consumer surplus. Since I = I = wxl/(x + s), we have the steady state welfare gains from trade equal to the gains in the consumer surplus from the differentiated good, which as we showed does not depend on the extent of the labor market frictions. 14 The only result here that requires the Pareto productivity assumption is the characterization of the change in the number of firms; without the Pareto assumption, M /M may be larger or smaller than H /H. Under the Pareto productivity distribution we can additionally obtain a closed-form solution for the production cutoff: θ d = [ ] f d 1 + (f d/f x ) k/(ε 1) 1 1/k τ k. f e [k/(ε 1) 1](r + δ) 14
Proposition 1 shows that the long-run welfare effects of a reduction in trade costs are not affected by the extent of the labor market rigidities and, in particular, do not depend on the hiring costs, b, and the level of the labor market tightness, x. The proportional change in the consumer surplus from the traded good, Q ζ, depends only on the change in the trade costs and the product market parameters, but does not depend on whether the labor market is frictionless or not. This is not to say that labor market frictions are inconsequential in this economy, but instead they only affect the overall level of economic activity, equally before and after the reduction in trade costs. This can be seen from the production cutoff condition (19), where Φ summarizes the effect of the labor market frictions on the level of output given the production cutoff, which as we show in the proof of Proposition 1 depends only on the product market parameters. Furthermore, household income in terms of the non-traded good, I, does not change with the trade costs. This is because in both steady states firms make zero profits on average, while the labor income stays unchanged, as both the employment rate and the wage rate stay the same. 15 The measure of the welfare gains used in our analysis (see (24)) is equal to the proportional change in the consumer surplus from the traded good, if trade does not affect the market income, as is the case when we compare two steady states. More generally, the welfare gains from trade also depend on the trade-induced change in the market income in the economy, as we discuss in detail in our analysis of the transition dynamics. We scale the measure of the gains from trade by the consumer surplus from the traded-good before the reduction in trade costs. We choose this measure, as it parallels the measure of welfare gains in a one-sector model and makes it invariant to the size of the non-traded sector, L. The sharp result of Proposition 1 on the irrelevance of the labor market frictions for the long-run welfare gains from trade is already intriguing, yet one may expect that most of the bite of the labor market frictions may occur during the transition dynamics. Indeed, labor market frictions may delay the increase in the traded-good output, Q, or result in transitional unemployment and a reduction in the market income during the transition. Furthermore, although the average employment per firm, H/M, stays constant across steady states, individual firms change their employment in different ways depending on their productivity. After the reduction in trade costs, the less productive non-exporting firms shrink, while the more productive exporting firms expand their employment (see the appendix). Labor market frictions slow down this reallocation, leading to misallocation of employment across firms, which 15 This is only the case with symmetric labor market frictions in both sectors, which is our benchmark. Outside this case, the sectoral unemployment rates are not the same, and the labor reallocation towards the traded sector resulting from trade liberalization changes the economy-wide employment rate (Helpman and Itskhoki, 2010). As a result, the labor market income is in general not constant and may go both up and down with the reduction in the trade costs. 15
may depress the welfare gains from trade along the transition. We now turn to the formal analysis of these issues. 4 Dynamic Gains from Trade We now study the dynamic transition of the economy from an initial steady state with a high variable trade cost, τ, to a new long-run equilibrium with a lower variable trade cost, τ < τ. The change in τ happens at t = 0 and is one-time, permanent and unexpected. We denote with a prime the new steady state, without a prime and no subscript the initial steady state, and with a time subscript the dynamic evolution of the variables. Before exploring the rich dynamic adjustment at the firm level with its impact on aggregate productivity and trade flows in the next section, we start our analysis here with a sharp analytical result characterizing the gains in the consumer surplus from trade the unique source of the long-run gains from trade, as we have shown in Propostion 1: Proposition 2 Along the transition path, Q t Q for all t 0, which holds with equality in all periods when there is entry of firms. Therefore, if there is entry in every period, the gains from trade in the consumer surplus, ( Q t /Q ) ζ ( Q /Q ) ζ for all t 0, are instantaneous, and, as follows from Proposition 1, independent from the labor market frictions. 16 Proof: First, from Lemma 1 and 3, we know that the labor market tightness, x, and the hiring cost, b, are constant in both sectors throughout the transition. This is achieved by the reallocation of unemployed from the homogenous to the differentiated sector in order to maintain the increased demand for labor in this sector. Assumption 2 ensures that there are enough unemployed economy-wide to satisfy this increased labor demand. This assumption imposes a joint upper bound on the relative size of the traded sector and the reduction in the trade costs, which we explore numerically in the following subsection. Second, we prove that there are two possibilities depending on the parameters of the model and the size of the trade liberalization: (i) Positive entry flow starting immediately at t = 0, M0 e > 0. In this case, Mt e > 0 for all t 0, Q 0 < Q and Q t = Q for all t > 0. That it, output jumps up on impact (with a one period lag) to the new long-run level. Since M e t > 0, the free entry condition (15) holds with equality for all t 0. Since b t = b and Q t = Q for all t > 0, every entrant that chooses to produce (i.e., with θ θ d ) does so in every period after entry and hence continuously hires workers to replace attrition. 16 Since entry into production happens with a one-period lag, which is required to hire workers, Proposition 2 formally applies for t, and by our convention we use an approximation t 0, as we consider small 0. 16
Therefore, the characterization of J V (θ) in Lemma 4 applies, with J V (θ) constant over time, and we can write the free entry condition as: θ [ t e (r+δ)( t t) max ι { ΦΘ t( ) } ] 1+β ι; θ fd ιf x, 0 d t dg(θ) = f e, (25) where we approximate a sum with small finite time increments with an integral. Taking the derivative with respect to time t, we have: θ max ι { ΦΘ ) } 1+β t( ι; θ fd ιf x, 0 dg(θ) = (r + δ)f e. Given the definition of Θ t in (7), the left-hand side of this equation is monotonically decreasing in Q t, and there exists a single value Q t = Q consistent with the above equality. This verifies our conjecture. (ii) No entry initially at t = 0, M e 0 = 0. In this case, there exists T e > 0 such that M e t = 0 and Q t > Q t+ Q for t [0, T e ), and Mt e > 0 and Q t = Q for t > T e. That is, output gradually declines (in finite time) to the new long-run level. Since b t = b for all t 0 and Q t is non-increasing, any entrant that chooses to produce at t > 0 also does so at all future dates. Similarly, every firm that chooses to export at t 0 also does so at all future dates. Therefore, every entrant that chooses to hire workers, continuously hires them in all future periods, either to replace attrition or also to expand employment in cases when Q t strictly declines. This again allows us to use the characterization of the value of an entrant from Lemma 4, however in this case the relationship in (25) holds initially as a strict inequality, and Mt e = 0 to satisfy complementary slackness. Due to firm death (as well as labor force attrition for some shrinking incumbents), during this initial period output Q t declines, until it reaches Q immediately after some finite time T e, at which the free entry condition starts to hold with equality and firms start to enter, M e t > 0 for all t T e. Finally, there are no other possibilities for the behavior of entry and output during the transition dynamics. Indeed, if Q 0 < Q, there is no future date t > 0 at which Q t > Q, i.e. overshoots the new long-run level. If such t > 0 existed, those firms that entered just before t would have made losses in expectation, violating the complementary slackness condition to the free entry condition (15). Furthermore, Q t necessarily jumps up to Q already at t =, as otherwise entrants at t = 0 are making positive profits in expectation, violating the free entry condition (15). Lastly, if Q 0 > Q, then there cannot exist a future date t > 0 such that Q t < Q. If such t existed, the firms could enter at t and make positive profits, violating the free entry condition (15). 17
Proposition 2 is a powerful result since it shows that all gains from trade in the consumer surplus the only source of the long-run gains from trade, as shown in Proposition 1 are realized immediately without delay and irrespectively of the extent of the labor market frictions, unless there is overshooting in the short run. The overshooting only happens in cases when there is no entry of firms in the short run an empirically unlikely scenario, which we explore numerically in the following sections. Of course, there is another sources of departure from the long-run gains from trade due to the after-shock transition specifically, the dynamics of the household market income, which we also quantify numerically below. However, it is only the net present value of this income that matters for welfare, and hence a representative family experiences the full amount of gains from trade instantaneously with no delay. The main force behind the result of Proposition 2 is the free entry condition of firms: both in the non-traded sector to ensure stable labor market tightness and hiring costs economy-wide, and more importantly in the traded sector to act as a buffer for product market competition. Indeed, the free entry condition pins down the single level of product market competition, Q t = Q, consistent with the firm entry (given the hiring cost b and trade cost τ ), and independently from the distribution of employment across incumbent firms. Indeed, the joint distribution of employment and productivity among the incumbents, G 0 (h, θ), which is a complex state variable in the model, is irrelevant for the result of the proposition, provided there is entry of new firms. The entry ceases only when there is no more slack left in the product market, i.e. Q t = Q for all t > 0. The lack of the dynamic effects of the labor market frictions on the consumer surplus from trade does not imply, however, that the labor market frictions are inconsequential for the allocation of resources and productivity after the reduction in trade cost, as we show next. 5 Dynamic Adjustment to Trade Proposition 2 requires no information on the micro-level allocation of employment across either incumbent firms or entrants, which masks the rich dynamic patterns of labor reallocation across firms within the traded sector, as well as across the two sectors. This firm-level reallocation, in turn, shapes the aggregate dynamics of employment, productivity and trade flows. In this section we provide a full characterization of this dynamic adjustment. While the model admits a full analytical characterization, it is convenient to illustrate the forces in the model by means of specific numerical examples which are representative of various possible dynamic patterns that arise for different parameter values. The qualitative dynamic patters are not sensitive to some of the parameters, and thus we choose to fix these 18
parameters at their conventional values. Table 1: Benchmark parameters Moment Parameter Value Comment Discount rate r 0.05 Exogenous separation rate s 0.2 s 0 = s Labor force attrition rate σ 0.175 Firm death rate δ 0.025 Job finding rate x 2 a 0 = a = 0.12 Relative elasticity of matching α 1 Unemployment benefit b u 0.4 Pareto shape parameter k 4 CES within sector ε 4 β = 3/4 Semi-elasticity across sectors 2 ζ = 1/2 Employment share in the traded sector 14% L = 10, f d = 0.05 Fraction of exitors 25% (r + δ)f e /f d = 2.7 Fraction of exporters 11% f x /f d = 1 Fraction of output exported 16% τ = 1.75 Trade liberalization τ = 1.375 Fraction of exporters 28% Fraction of output exported 28% Notes: All rates are annualized: for example, the separation rate is 20% per year and the job finding rate of 2 corresponds to an average unemployment duration of 1/x = 0.5 years. α = 1 is the ratio of the two elasticities of the matching function with respect to employment and vacancies respectively, which sum to one ensuring constant returns to scale in matching. The unemployment benefit b u = 0.4 corresponds to a 45% replacement ratio. Semi-elasticity across sectors is equal to 1/(1 ζ) = 2. The Pareto shape parameter for employment and sales in steady state is equal to k/(ε 1) = 1.33. The middle panel shows the values of fixed costs f d, f d, f e and iceberg trade cost τ used to match the moments on exports and exit; fraction of exitors is the fraction of entrants that choose not to produce in the initial steady state. Table 1 summarizes the benchmark values of parameters and corresponding empirical moments. 17 We calibrate the productivity of the matching function a key parameter controlling the extent of labor market frictions to match the average unemployment duration of 6 month, corresponding to an annualized job finding rate of x = 2. This is more characteristic of the slower European rather than the more dynamic US labor market, and corresponds to an economy-wide unemployment rate of 9%. In addition, we show the sensitivity of the results to a wide range of variation in this parameter. 18 17 See the notes to Table 1 for additional details about the calibration. 18 More generally, the labor market frictions in this model are fully captured by a vector (x, b, b u, s), where 19